Managers-Net

Linear and quadratic equations

Their role in management services

Linear and quadratic equations are useful in management as mathematical models which describe any
situation and subsequently can be used for predicting outcomes and trends.
As models they can produce rapid solutions to defined situations once the parameters of that situation
have been inserted into the model.

Linear equations

It is instructive to see the anatomy of a simple linear equation. Y = ax + b is an example.

This equation has one variable, x, which can take any value, and two constants,
a and b which are fixed values in this model. The value Y is the answer.

Example

Consider the following linear model that describes the time necessary to load a van
with parcels for distribution.

time to load the van = (a x number of parcels) + b

where a represents the time per parcel and b is a constant value representing the time
for ancillary parts of the job, for example, cleaning out the van each time ready for the next load.

As a result of timing the loading of the van several times for different loads, this model,
arrived at by regression analysis,
has calculated values for "a" of 0.3 minutes and "b" of 3.4 minutes.
(Note: the reader is referred to the paper on "Multiple regression analysis"
in this series, on this managers-net website.)
Replacing "a" and "b" with these values results in a useable model of:

time to load the van = (0.3 x number of parcels) + 3.4

As an example suppose we have 12 parcels to deliver.
Inserting this figure in our model, it should take:
(0.3 x 12) + 3.4 or 7.0 minutes to load the van.
The time increases in straight proportion to the number of parcels loaded, in other words a graph of this
model would produce a straight line starting at 3.4 minutes and rising as the number of parcels increases.

This type of model has many uses in addition to work measurement, where there is only one variable to consider.

Quadratic equations

In contrast to the above, a situation having one variable might not be linear in its behaviour in
which case a quadratic equation model may describe it more accurately. This takes the general form:

Y = ax2 + bx + c, where a, b and c are constants and x is the variable.

This model can be used to represent certain situations that are not linear. As x increases,
the result, Y, increases more rapidly than it would with direct proportion, illustrated previously.